Equation Simplification in Mathematica

Question

I wonder if the following equation could be further simplified by Mathematica. There are $22$ Integrals involved and many of them are in the range $y_l$ and $y_u$. I would guess that at least those terms could be collected in the same integral. I used the command FullSimplify but the result is the same with the input. Are there some more clever ways to simplify such type of equations?

(2*(1 - \[Epsilon]0)^2*Integrate[((Sqrt[f1[y]*l[yu]] - Sqrt[f0[y]*l[yl]
*l[yu]])*(-Sqrt[f1[y]*l[yl]] + Sqrt[f0[y]*l[yl]*l[yu]]))/(-Sqrt[l[yl]] + 
Sqrt[l[yu]])^2, {y, yl, yu}] - 2*(Integrate[(Sqrt[f0[y]]*(-Sqrt[f1[y]*l[yl]] + 
Sqrt[f0[y]*l[yl]*l[yu]]))/(-Sqrt[l[yl]] + Sqrt[l[yu]]), {y, yl, yu}] +
 Integrate[f0[y], {y, -Infinity, yl}]*Sqrt[l[yl]])*(Integrate[(Sqrt[f0[y]]*
 (Sqrt[f1[y]*l[yu]] - Sqrt[f0[y]*l[yl]*l[yu]]))/(-Sqrt[l[yl]] + Sqrt[l[yu]]), 
{y, yl, yu}] + Integrate[f0[y], {y, yu, Infinity}]*Sqrt[l[yu]]) -
 Sqrt[(-2*(1 - \[Epsilon]0)^2*Integrate[((Sqrt[f1[y]*l[yu]] - 
Sqrt[f0[y]*l[yl]*l[yu]])*(-Sqrt[f1[y]*l[yl]] +
 Sqrt[f0[y]*l[yl]*l[yu]]))/(-Sqrt[l[yl]] + Sqrt[l[yu]])^2, {y, yl, yu}] + 
2*(Integrate[(Sqrt[f0[y]]*(-Sqrt[f1[y]*l[yl]] + 
Sqrt[f0[y]*l[yl]*l[yu]]))/(-Sqrt[l[yl]] + Sqrt[l[yu]]), {y, yl, yu}] + 
Integrate[f0[y], {y, -Infinity, yl}]*Sqrt[l[yl]])*(Integrate[(Sqrt[f0[y]]*
(Sqrt[f1[y]*l[yu]] - Sqrt[f0[y]*l[yl]*l[yu]]))/(-Sqrt[l[yl]] + 
Sqrt[l[yu]]), {y, yl, yu}] + Integrate[f0[y], {y, yu, Infinity}]*
Sqrt[l[yu]]))^2 - 4*((Integrate[(Sqrt[f0[y]]*(-Sqrt[f1[y]*l[yl]] + 
Sqrt[f0[y]*l[yl]*l[yu]]))/(-Sqrt[l[yl]] + Sqrt[l[yu]]), {y, yl, yu}] +
 Integrate[f0[y], {y, -Infinity, yl}]*Sqrt[l[yl]])^2 - 
(1 -\[Epsilon]0)^2*(Integrate[(Sqrt[f0[y]]*(-Sqrt[f1[y]*l[yl]] + 
Sqrt[f0[y]*l[yl]*l[yu]])^2)/(-Sqrt[l[yl]] + Sqrt[l[yu]])^2, {y, yl, yu}] + 
Integrate[f0[y], {y, -Infinity, yl}]*l[yl]))*((Integrate[(Sqrt[f0[y]]*
(Sqrt[f1[y]*l[yu]] - Sqrt[f0[y]*l[yl]*l[yu]]))/(-Sqrt[l[yl]] + 
Sqrt[l[yu]]), {y, yl, yu}] + Integrate[f0[y], {y, yu, Infinity}]*
Sqrt[l[yu]])^2 - (1 - \[Epsilon]0)^2*(Integrate[(Sqrt[f1[y]*l[yu]] - 
Sqrt[f0[y]*l[yl]*l[yu]])^2/(-Sqrt[l[yl]] + Sqrt[l[yu]])^2, {y, yl, yu}] + 
Integrate[f0[y], {y, yu, Infinity}]*l[yu]))])/(2*((Integrate[(Sqrt[f0[y]]*
(Sqrt[f1[y]*l[yu]] - Sqrt[f0[y]*l[yl]*l[yu]]))/(-Sqrt[l[yl]] + Sqrt[l[yu]]), 
{y, yl, yu}] + Integrate[f0[y], {y, yu, Infinity}]*Sqrt[l[yu]])^2 - 
(1 - \[Epsilon]0)^2*(Integrate[(Sqrt[f1[y]*l[yu]] - 
Sqrt[f0[y]*l[yl]*l[yu]])^2/(-Sqrt[l[yl]] + Sqrt[l[yu]])^2, {y, yl, yu}] 
+Integrate[f0[y], {y, yu, Infinity}]*l[yu])))

Answer 1

Let's say your expression is expr. Then you can parse this this way, first let's take a look at a structure of the expr, in order to do this we can create a list of substititutions. Of course we are doing this because we have seen that there are common terms involving integrals:

 (subs = With[{set = Cases[expr, _Integrate, ∞] // DeleteDuplicates},
           Table[set[[i]] :> Evaluate@int[i], {i, Length@set}]]

  ) // Grid[{#}\[Transpose], Alignment -> Right, BaseStyle -> {18}] &

expr /. subs

If you take a closer look you can see that there are no terms that could be collected inside common integral.

Even if there would be it can be tough according to Why aren't these additions of integrals and summations equal?

So as you can see FullSimplify is not enough to do this. (in the case where it can be done) Distributing Integrate is quite easy and is shown in the linked answer. If you want to gather terms to the common Integrate you can try with the following pattern:

5 Integrate[f[x], {x, x1, x2}] - 2 Integrate[g[x], {x, x1, x2}] //. (
  con1_. Integrate[a_, {x_, l__}] + con2_. Integrate[b_, {x_, l__}]) :> 
  Integrate[con1 a + con2 b, {x, l}] /; D[con1, x] == 0 && D[con2, x] == 0